Consider the following cases :
1).Peter and his brother came to my house.
We cannot use si because of rule (b) - it is part of the subject and so
Petro kaj lia frato venis al mia domo (E: fraternal)
2).Peter begged George to speak to his brother
In English it is not clear whose brother (Peter's or George's). In Esperanto we cannot use sia unless it refers to the subject- but what is the subject? Peter is subject of the verb 'begged' whereas George is the object of 'begged' and at the same time the subject of 'to speak'.
Petro petis Georgon paroli al sia frato
Here the sia refers to the subject of its clause i.e. the subject of the verb paroli. Thus it refers to George - even though it is in the accusative being the object of the previous verb petis.
So this means Peter begged George to talk to George's brother
Petro petis Georgon paroli al lia frato
Lia cannot refer to the subject George and so must refer to some other noun (Peter). Here the meaning is Peter begged George to talk to Peter's brother.
3) Consider this apparent difficulty :
La homoj elektis lin ilia prezidanto (The men elected him their president) (E : Homo sapiens)
Does not 'their' refer back to the subject 'the men'?
Careful analysis will show that ilia prezidanto to be in the nominative (-o ending). It cannot be the object, therfore, of elektis. The sentence really is 'The men elected him (to be) their president and so there are two verbs elektis and the understood esti.
Thus ilia prezidanto refers to the subject of the understood verb 'to be' i.e. to lin and not to the subject of a different clause with a different verb (i.e. la homoj and elektis) and so sia must not be used.
The exact use of the reflexive in the last few examples is advanced and need not trouble the beginner too much - even experts can get into trouble with this! It is only given here for completeness.
6). Mine
Mine is used in English to replace the personal adjective and a noun e.g.
Here is his book - No, it's mine (i.e. my book)